Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256)
gp: K = bnfinit(y^32 - 8*y^31 + 64*y^30 - 332*y^29 + 1604*y^28 - 6256*y^27 + 22310*y^26 - 68580*y^25 + 189272*y^24 - 457172*y^23 + 954788*y^22 - 1681104*y^21 + 2211865*y^20 - 1508956*y^19 - 2584912*y^18 + 11838608*y^17 - 24128969*y^16 + 31887592*y^15 - 15354096*y^14 - 32328344*y^13 + 86966400*y^12 - 105636528*y^11 + 18312240*y^10 + 107553536*y^9 - 98914352*y^8 + 2393408*y^7 + 40426432*y^6 - 24013312*y^5 + 5360768*y^4 - 207104*y^3 - 36096*y^2 + 1536*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256)
\( x^{32} - 8 x^{31} + 64 x^{30} - 332 x^{29} + 1604 x^{28} - 6256 x^{27} + 22310 x^{26} - 68580 x^{25} + \cdots + 256 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1119917091733566341117149537716450109685760000000000000000\)
\(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 1049^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.64\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}5^{1/2}29^{1/2}1049^{1/2}\approx 1910.6334028274498$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(29\), \(1049\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{20}-\frac{1}{4}a^{18}-\frac{1}{4}a^{14}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{21}-\frac{1}{4}a^{19}-\frac{1}{4}a^{15}+\frac{1}{8}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{22}-\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{8}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{1}{16}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{16}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{16}a^{23}-\frac{1}{4}a^{19}-\frac{1}{8}a^{17}-\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{16}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{16}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{64}a^{24}-\frac{1}{16}a^{21}+\frac{3}{32}a^{18}-\frac{1}{16}a^{17}+\frac{1}{8}a^{16}-\frac{1}{16}a^{15}+\frac{3}{16}a^{14}-\frac{7}{64}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{13}{64}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{16}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{64}a^{25}-\frac{5}{32}a^{19}+\frac{3}{16}a^{18}+\frac{1}{8}a^{17}-\frac{3}{16}a^{16}-\frac{1}{16}a^{15}+\frac{9}{64}a^{13}+\frac{3}{16}a^{12}-\frac{1}{16}a^{10}-\frac{29}{64}a^{9}-\frac{1}{4}a^{7}+\frac{7}{16}a^{6}-\frac{7}{16}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{128}a^{26}-\frac{1}{32}a^{23}+\frac{3}{64}a^{20}-\frac{1}{32}a^{19}-\frac{3}{16}a^{18}+\frac{7}{32}a^{17}+\frac{3}{32}a^{16}+\frac{25}{128}a^{14}+\frac{7}{32}a^{13}+\frac{3}{16}a^{11}+\frac{19}{128}a^{10}+\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{15}{32}a^{6}+\frac{3}{8}a^{4}-\frac{3}{8}a^{2}$, $\frac{1}{128}a^{27}+\frac{3}{64}a^{21}-\frac{1}{32}a^{20}+\frac{1}{16}a^{19}-\frac{3}{32}a^{18}-\frac{1}{32}a^{17}-\frac{1}{4}a^{16}-\frac{23}{128}a^{15}+\frac{3}{32}a^{14}-\frac{1}{32}a^{12}+\frac{3}{128}a^{11}+\frac{3}{32}a^{8}+\frac{9}{32}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{512}a^{28}-\frac{1}{256}a^{26}-\frac{1}{128}a^{25}-\frac{1}{128}a^{24}+\frac{1}{64}a^{23}+\frac{3}{256}a^{22}+\frac{3}{128}a^{21}+\frac{7}{128}a^{20}+\frac{9}{128}a^{19}-\frac{23}{128}a^{18}+\frac{3}{64}a^{17}-\frac{31}{512}a^{16}+\frac{11}{128}a^{15}+\frac{15}{256}a^{14}-\frac{1}{16}a^{13}-\frac{17}{512}a^{12}-\frac{3}{16}a^{11}-\frac{11}{256}a^{10}-\frac{7}{16}a^{9}-\frac{29}{64}a^{8}+\frac{1}{4}a^{7}+\frac{21}{64}a^{6}+\frac{1}{4}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{2}+\frac{3}{8}$, $\frac{1}{512}a^{29}-\frac{1}{256}a^{27}-\frac{1}{128}a^{25}-\frac{5}{256}a^{23}+\frac{3}{128}a^{22}-\frac{1}{128}a^{21}-\frac{1}{128}a^{20}+\frac{5}{128}a^{19}+\frac{1}{64}a^{18}+\frac{113}{512}a^{17}+\frac{7}{128}a^{16}-\frac{33}{256}a^{15}+\frac{25}{128}a^{14}+\frac{95}{512}a^{13}-\frac{5}{64}a^{12}+\frac{53}{256}a^{11}+\frac{27}{128}a^{10}-\frac{29}{64}a^{9}+\frac{29}{64}a^{8}-\frac{27}{64}a^{7}+\frac{9}{32}a^{6}-\frac{7}{16}a^{5}-\frac{1}{16}a^{4}-\frac{7}{16}a^{3}-\frac{3}{8}a^{2}-\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{42\!\cdots\!36}a^{30}+\frac{14\!\cdots\!79}{21\!\cdots\!68}a^{29}-\frac{16\!\cdots\!39}{19\!\cdots\!88}a^{28}-\frac{44\!\cdots\!49}{16\!\cdots\!56}a^{27}+\frac{59\!\cdots\!77}{10\!\cdots\!84}a^{26}-\frac{49\!\cdots\!13}{12\!\cdots\!68}a^{25}+\frac{48\!\cdots\!03}{21\!\cdots\!68}a^{24}+\frac{37\!\cdots\!91}{26\!\cdots\!96}a^{23}-\frac{37\!\cdots\!87}{10\!\cdots\!84}a^{22}-\frac{22\!\cdots\!35}{15\!\cdots\!52}a^{21}-\frac{33\!\cdots\!13}{10\!\cdots\!84}a^{20}-\frac{76\!\cdots\!51}{26\!\cdots\!96}a^{19}-\frac{65\!\cdots\!15}{42\!\cdots\!36}a^{18}-\frac{15\!\cdots\!83}{21\!\cdots\!68}a^{17}-\frac{19\!\cdots\!73}{21\!\cdots\!68}a^{16}-\frac{83\!\cdots\!05}{34\!\cdots\!72}a^{15}-\frac{45\!\cdots\!11}{38\!\cdots\!76}a^{14}-\frac{78\!\cdots\!13}{19\!\cdots\!88}a^{13}-\frac{94\!\cdots\!27}{21\!\cdots\!68}a^{12}-\frac{17\!\cdots\!61}{10\!\cdots\!84}a^{11}+\frac{11\!\cdots\!25}{52\!\cdots\!92}a^{10}+\frac{14\!\cdots\!97}{13\!\cdots\!48}a^{9}+\frac{90\!\cdots\!37}{52\!\cdots\!92}a^{8}+\frac{16\!\cdots\!51}{26\!\cdots\!96}a^{7}+\frac{42\!\cdots\!83}{13\!\cdots\!48}a^{6}-\frac{15\!\cdots\!55}{33\!\cdots\!12}a^{5}-\frac{14\!\cdots\!67}{13\!\cdots\!48}a^{4}+\frac{29\!\cdots\!11}{66\!\cdots\!24}a^{3}-\frac{13\!\cdots\!47}{60\!\cdots\!84}a^{2}+\frac{15\!\cdots\!43}{33\!\cdots\!12}a-\frac{35\!\cdots\!73}{82\!\cdots\!78}$, $\frac{1}{29\!\cdots\!76}a^{31}-\frac{24\!\cdots\!73}{36\!\cdots\!72}a^{30}+\frac{44\!\cdots\!31}{92\!\cdots\!68}a^{29}-\frac{11\!\cdots\!53}{73\!\cdots\!44}a^{28}+\frac{46\!\cdots\!47}{18\!\cdots\!36}a^{27}+\frac{12\!\cdots\!69}{36\!\cdots\!72}a^{26}+\frac{37\!\cdots\!87}{14\!\cdots\!88}a^{25}-\frac{55\!\cdots\!77}{73\!\cdots\!44}a^{24}+\frac{10\!\cdots\!21}{36\!\cdots\!72}a^{23}-\frac{13\!\cdots\!85}{73\!\cdots\!44}a^{22}+\frac{28\!\cdots\!03}{73\!\cdots\!44}a^{21}+\frac{60\!\cdots\!67}{16\!\cdots\!76}a^{20}-\frac{58\!\cdots\!43}{29\!\cdots\!76}a^{19}+\frac{13\!\cdots\!53}{73\!\cdots\!44}a^{18}+\frac{11\!\cdots\!83}{92\!\cdots\!68}a^{17}+\frac{59\!\cdots\!53}{36\!\cdots\!72}a^{16}-\frac{61\!\cdots\!99}{26\!\cdots\!16}a^{15}-\frac{78\!\cdots\!11}{16\!\cdots\!76}a^{14}+\frac{14\!\cdots\!83}{36\!\cdots\!72}a^{13}+\frac{40\!\cdots\!03}{16\!\cdots\!76}a^{12}+\frac{38\!\cdots\!25}{73\!\cdots\!44}a^{11}-\frac{51\!\cdots\!61}{36\!\cdots\!72}a^{10}+\frac{74\!\cdots\!21}{36\!\cdots\!72}a^{9}+\frac{22\!\cdots\!67}{92\!\cdots\!68}a^{8}+\frac{75\!\cdots\!75}{16\!\cdots\!76}a^{7}-\frac{40\!\cdots\!85}{92\!\cdots\!68}a^{6}+\frac{19\!\cdots\!27}{83\!\cdots\!88}a^{5}-\frac{28\!\cdots\!81}{23\!\cdots\!92}a^{4}-\frac{76\!\cdots\!47}{23\!\cdots\!92}a^{3}+\frac{12\!\cdots\!75}{23\!\cdots\!92}a^{2}-\frac{46\!\cdots\!99}{23\!\cdots\!92}a-\frac{65\!\cdots\!45}{57\!\cdots\!98}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{276}$, which has order $552$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $552$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{6550795017700884610869107519}{66089696803178893816510666362224} a^{31} - \frac{98697440185561635762474650281}{132179393606357787633021332724448} a^{30} + \frac{790369874190999902565619079915}{132179393606357787633021332724448} a^{29} - \frac{1982179212780856506366972297353}{66089696803178893816510666362224} a^{28} + \frac{9516455585906551027582959490831}{66089696803178893816510666362224} a^{27} - \frac{36199060207996033224367059255425}{66089696803178893816510666362224} a^{26} + \frac{7975387185718478354351572222905}{4130606050198680863531916647639} a^{25} - \frac{191640835149922642527446373097901}{33044848401589446908255333181112} a^{24} + \frac{1038138011372345119944480153982503}{66089696803178893816510666362224} a^{23} - \frac{55457825233402976158744650307597}{1502038563708611223102515144596} a^{22} + \frac{307644297157732482724549749285763}{4130606050198680863531916647639} a^{21} - \frac{4124334649574668655444207996695067}{33044848401589446908255333181112} a^{20} + \frac{9667149718303476120696155410877437}{66089696803178893816510666362224} a^{19} - \frac{7330198557959494938531140609550941}{132179393606357787633021332724448} a^{18} - \frac{41757271039455506275631968510074849}{132179393606357787633021332724448} a^{17} + \frac{69301951037342699044646758258258593}{66089696803178893816510666362224} a^{16} - \frac{5591366791236275060679985503007731}{3004077127417222446205030289192} a^{15} + \frac{25334272064266043645939966569751365}{12016308509668889784820121156768} a^{14} - \frac{22003141796883143394431077108945091}{132179393606357787633021332724448} a^{13} - \frac{124553039801555958513862702624761083}{33044848401589446908255333181112} a^{12} + \frac{468354182422226444096089631968293263}{66089696803178893816510666362224} a^{11} - \frac{439628870984999979627875885694779427}{66089696803178893816510666362224} a^{10} - \frac{43473349055882455359682668176499947}{16522424200794723454127666590556} a^{9} + \frac{66339814844812699241680999962609529}{6008154254834444892410060578384} a^{8} - \frac{80779735703397496909287529921116365}{16522424200794723454127666590556} a^{7} - \frac{61630775492931854431962795770415935}{16522424200794723454127666590556} a^{6} + \frac{15505709329481119420195420395194001}{4130606050198680863531916647639} a^{5} - \frac{5108685491154896364381271731194839}{8261212100397361727063833295278} a^{4} - \frac{1585391524389208929994308439366725}{4130606050198680863531916647639} a^{3} + \frac{636829934720720247551711015815083}{4130606050198680863531916647639} a^{2} - \frac{308948878056587682740398880962}{61650836570129565127342039517} a - \frac{836225992184205854615977913422}{4130606050198680863531916647639} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{17\!\cdots\!25}{58\!\cdots\!08}a^{31}-\frac{27\!\cdots\!47}{11\!\cdots\!16}a^{30}+\frac{54\!\cdots\!69}{29\!\cdots\!04}a^{29}-\frac{56\!\cdots\!53}{58\!\cdots\!08}a^{28}+\frac{13\!\cdots\!59}{29\!\cdots\!04}a^{27}-\frac{53\!\cdots\!73}{29\!\cdots\!04}a^{26}+\frac{19\!\cdots\!57}{29\!\cdots\!04}a^{25}-\frac{11\!\cdots\!33}{58\!\cdots\!08}a^{24}+\frac{16\!\cdots\!49}{29\!\cdots\!04}a^{23}-\frac{35\!\cdots\!57}{26\!\cdots\!64}a^{22}+\frac{81\!\cdots\!95}{29\!\cdots\!04}a^{21}-\frac{14\!\cdots\!31}{29\!\cdots\!04}a^{20}+\frac{37\!\cdots\!65}{58\!\cdots\!08}a^{19}-\frac{50\!\cdots\!55}{11\!\cdots\!16}a^{18}-\frac{13\!\cdots\!47}{18\!\cdots\!44}a^{17}+\frac{20\!\cdots\!75}{58\!\cdots\!08}a^{16}-\frac{37\!\cdots\!71}{53\!\cdots\!28}a^{15}+\frac{98\!\cdots\!97}{10\!\cdots\!56}a^{14}-\frac{12\!\cdots\!17}{29\!\cdots\!04}a^{13}-\frac{55\!\cdots\!99}{58\!\cdots\!08}a^{12}+\frac{36\!\cdots\!81}{14\!\cdots\!52}a^{11}-\frac{44\!\cdots\!49}{14\!\cdots\!52}a^{10}+\frac{36\!\cdots\!45}{72\!\cdots\!76}a^{9}+\frac{41\!\cdots\!23}{13\!\cdots\!32}a^{8}-\frac{10\!\cdots\!13}{36\!\cdots\!88}a^{7}+\frac{17\!\cdots\!23}{36\!\cdots\!88}a^{6}+\frac{20\!\cdots\!21}{18\!\cdots\!44}a^{5}-\frac{24\!\cdots\!47}{36\!\cdots\!88}a^{4}+\frac{73\!\cdots\!23}{45\!\cdots\!36}a^{3}-\frac{24\!\cdots\!37}{18\!\cdots\!44}a^{2}+\frac{85\!\cdots\!33}{11\!\cdots\!09}a-\frac{75\!\cdots\!61}{11\!\cdots\!09}$, $\frac{10\!\cdots\!53}{44\!\cdots\!12}a^{31}-\frac{97\!\cdots\!05}{44\!\cdots\!12}a^{30}+\frac{77\!\cdots\!69}{44\!\cdots\!12}a^{29}-\frac{42\!\cdots\!07}{44\!\cdots\!12}a^{28}+\frac{10\!\cdots\!57}{22\!\cdots\!56}a^{27}-\frac{42\!\cdots\!67}{22\!\cdots\!56}a^{26}+\frac{15\!\cdots\!83}{22\!\cdots\!56}a^{25}-\frac{48\!\cdots\!31}{22\!\cdots\!56}a^{24}+\frac{13\!\cdots\!85}{22\!\cdots\!56}a^{23}-\frac{34\!\cdots\!81}{22\!\cdots\!56}a^{22}+\frac{18\!\cdots\!91}{55\!\cdots\!64}a^{21}-\frac{68\!\cdots\!67}{11\!\cdots\!28}a^{20}+\frac{39\!\cdots\!09}{44\!\cdots\!12}a^{19}-\frac{35\!\cdots\!21}{44\!\cdots\!12}a^{18}-\frac{16\!\cdots\!73}{40\!\cdots\!92}a^{17}+\frac{16\!\cdots\!01}{44\!\cdots\!12}a^{16}-\frac{34\!\cdots\!21}{40\!\cdots\!92}a^{15}+\frac{51\!\cdots\!93}{40\!\cdots\!92}a^{14}-\frac{43\!\cdots\!57}{44\!\cdots\!12}a^{13}-\frac{28\!\cdots\!69}{44\!\cdots\!12}a^{12}+\frac{67\!\cdots\!35}{22\!\cdots\!56}a^{11}-\frac{98\!\cdots\!49}{22\!\cdots\!56}a^{10}+\frac{66\!\cdots\!49}{27\!\cdots\!32}a^{9}+\frac{82\!\cdots\!97}{27\!\cdots\!32}a^{8}-\frac{28\!\cdots\!61}{55\!\cdots\!64}a^{7}+\frac{84\!\cdots\!47}{55\!\cdots\!64}a^{6}+\frac{11\!\cdots\!31}{69\!\cdots\!08}a^{5}-\frac{11\!\cdots\!34}{79\!\cdots\!91}a^{4}+\frac{54\!\cdots\!57}{13\!\cdots\!16}a^{3}-\frac{13\!\cdots\!19}{13\!\cdots\!16}a^{2}-\frac{40\!\cdots\!43}{69\!\cdots\!08}a-\frac{56\!\cdots\!23}{69\!\cdots\!08}$, $\frac{45\!\cdots\!01}{27\!\cdots\!32}a^{31}-\frac{54\!\cdots\!43}{44\!\cdots\!12}a^{30}+\frac{44\!\cdots\!29}{44\!\cdots\!12}a^{29}-\frac{22\!\cdots\!41}{44\!\cdots\!12}a^{28}+\frac{53\!\cdots\!01}{22\!\cdots\!56}a^{27}-\frac{20\!\cdots\!61}{22\!\cdots\!56}a^{26}+\frac{22\!\cdots\!35}{69\!\cdots\!08}a^{25}-\frac{22\!\cdots\!91}{22\!\cdots\!56}a^{24}+\frac{61\!\cdots\!01}{22\!\cdots\!56}a^{23}-\frac{14\!\cdots\!21}{22\!\cdots\!56}a^{22}+\frac{15\!\cdots\!31}{11\!\cdots\!28}a^{21}-\frac{26\!\cdots\!91}{11\!\cdots\!28}a^{20}+\frac{52\!\cdots\!83}{17\!\cdots\!02}a^{19}-\frac{86\!\cdots\!59}{44\!\cdots\!12}a^{18}-\frac{16\!\cdots\!97}{40\!\cdots\!92}a^{17}+\frac{75\!\cdots\!91}{44\!\cdots\!12}a^{16}-\frac{67\!\cdots\!61}{20\!\cdots\!96}a^{15}+\frac{17\!\cdots\!75}{40\!\cdots\!92}a^{14}-\frac{79\!\cdots\!25}{44\!\cdots\!12}a^{13}-\frac{20\!\cdots\!15}{44\!\cdots\!12}a^{12}+\frac{26\!\cdots\!49}{22\!\cdots\!56}a^{11}-\frac{31\!\cdots\!27}{22\!\cdots\!56}a^{10}+\frac{67\!\cdots\!37}{55\!\cdots\!64}a^{9}+\frac{75\!\cdots\!19}{55\!\cdots\!64}a^{8}-\frac{62\!\cdots\!27}{55\!\cdots\!64}a^{7}+\frac{10\!\cdots\!45}{55\!\cdots\!64}a^{6}+\frac{58\!\cdots\!09}{13\!\cdots\!16}a^{5}-\frac{48\!\cdots\!91}{12\!\cdots\!56}a^{4}+\frac{15\!\cdots\!73}{13\!\cdots\!16}a^{3}-\frac{41\!\cdots\!89}{13\!\cdots\!16}a^{2}-\frac{11\!\cdots\!05}{69\!\cdots\!08}a+\frac{40\!\cdots\!49}{69\!\cdots\!08}$, $\frac{55\!\cdots\!51}{73\!\cdots\!44}a^{31}-\frac{43\!\cdots\!95}{73\!\cdots\!44}a^{30}+\frac{17\!\cdots\!37}{36\!\cdots\!72}a^{29}-\frac{17\!\cdots\!95}{73\!\cdots\!44}a^{28}+\frac{13\!\cdots\!49}{11\!\cdots\!96}a^{27}-\frac{39\!\cdots\!07}{85\!\cdots\!12}a^{26}+\frac{59\!\cdots\!91}{36\!\cdots\!72}a^{25}-\frac{18\!\cdots\!65}{36\!\cdots\!72}a^{24}+\frac{12\!\cdots\!83}{92\!\cdots\!68}a^{23}-\frac{18\!\cdots\!29}{54\!\cdots\!16}a^{22}+\frac{12\!\cdots\!05}{18\!\cdots\!36}a^{21}-\frac{25\!\cdots\!43}{20\!\cdots\!72}a^{20}+\frac{11\!\cdots\!43}{73\!\cdots\!44}a^{19}-\frac{73\!\cdots\!23}{73\!\cdots\!44}a^{18}-\frac{74\!\cdots\!33}{36\!\cdots\!72}a^{17}+\frac{63\!\cdots\!65}{73\!\cdots\!44}a^{16}-\frac{11\!\cdots\!53}{66\!\cdots\!04}a^{15}+\frac{14\!\cdots\!23}{66\!\cdots\!04}a^{14}-\frac{35\!\cdots\!93}{36\!\cdots\!72}a^{13}-\frac{16\!\cdots\!23}{66\!\cdots\!04}a^{12}+\frac{57\!\cdots\!39}{92\!\cdots\!68}a^{11}-\frac{27\!\cdots\!97}{36\!\cdots\!72}a^{10}+\frac{33\!\cdots\!33}{46\!\cdots\!84}a^{9}+\frac{74\!\cdots\!43}{92\!\cdots\!68}a^{8}-\frac{34\!\cdots\!55}{52\!\cdots\!18}a^{7}-\frac{34\!\cdots\!81}{92\!\cdots\!68}a^{6}+\frac{77\!\cdots\!59}{26\!\cdots\!09}a^{5}-\frac{35\!\cdots\!09}{23\!\cdots\!92}a^{4}+\frac{31\!\cdots\!47}{11\!\cdots\!96}a^{3}+\frac{14\!\cdots\!37}{23\!\cdots\!92}a^{2}-\frac{10\!\cdots\!91}{57\!\cdots\!98}a-\frac{19\!\cdots\!23}{17\!\cdots\!88}$, $\frac{16\!\cdots\!69}{14\!\cdots\!88}a^{31}-\frac{26\!\cdots\!87}{29\!\cdots\!76}a^{30}+\frac{26\!\cdots\!25}{36\!\cdots\!72}a^{29}-\frac{27\!\cdots\!55}{73\!\cdots\!44}a^{28}+\frac{13\!\cdots\!49}{73\!\cdots\!44}a^{27}-\frac{12\!\cdots\!13}{18\!\cdots\!36}a^{26}+\frac{18\!\cdots\!03}{73\!\cdots\!44}a^{25}-\frac{11\!\cdots\!45}{14\!\cdots\!88}a^{24}+\frac{15\!\cdots\!67}{73\!\cdots\!44}a^{23}-\frac{18\!\cdots\!95}{36\!\cdots\!72}a^{22}+\frac{78\!\cdots\!97}{73\!\cdots\!44}a^{21}-\frac{13\!\cdots\!61}{73\!\cdots\!44}a^{20}+\frac{36\!\cdots\!09}{14\!\cdots\!88}a^{19}-\frac{48\!\cdots\!07}{29\!\cdots\!76}a^{18}-\frac{21\!\cdots\!11}{73\!\cdots\!44}a^{17}+\frac{97\!\cdots\!55}{73\!\cdots\!44}a^{16}-\frac{35\!\cdots\!11}{13\!\cdots\!08}a^{15}+\frac{94\!\cdots\!65}{26\!\cdots\!16}a^{14}-\frac{61\!\cdots\!01}{36\!\cdots\!72}a^{13}-\frac{26\!\cdots\!97}{73\!\cdots\!44}a^{12}+\frac{17\!\cdots\!47}{18\!\cdots\!36}a^{11}-\frac{86\!\cdots\!55}{73\!\cdots\!44}a^{10}+\frac{17\!\cdots\!21}{92\!\cdots\!68}a^{9}+\frac{44\!\cdots\!59}{36\!\cdots\!72}a^{8}-\frac{50\!\cdots\!49}{46\!\cdots\!84}a^{7}+\frac{28\!\cdots\!25}{18\!\cdots\!36}a^{6}+\frac{51\!\cdots\!55}{11\!\cdots\!96}a^{5}-\frac{24\!\cdots\!53}{92\!\cdots\!68}a^{4}+\frac{13\!\cdots\!41}{23\!\cdots\!92}a^{3}-\frac{55\!\cdots\!09}{23\!\cdots\!92}a^{2}-\frac{11\!\cdots\!27}{57\!\cdots\!98}a+\frac{17\!\cdots\!89}{23\!\cdots\!92}$, $\frac{24\!\cdots\!21}{29\!\cdots\!76}a^{31}-\frac{98\!\cdots\!93}{14\!\cdots\!88}a^{30}+\frac{98\!\cdots\!61}{18\!\cdots\!36}a^{29}-\frac{20\!\cdots\!89}{73\!\cdots\!44}a^{28}+\frac{49\!\cdots\!67}{36\!\cdots\!72}a^{27}-\frac{19\!\cdots\!55}{36\!\cdots\!72}a^{26}+\frac{27\!\cdots\!07}{14\!\cdots\!88}a^{25}-\frac{10\!\cdots\!19}{18\!\cdots\!36}a^{24}+\frac{29\!\cdots\!87}{18\!\cdots\!36}a^{23}-\frac{28\!\cdots\!61}{73\!\cdots\!44}a^{22}+\frac{58\!\cdots\!49}{73\!\cdots\!44}a^{21}-\frac{51\!\cdots\!13}{36\!\cdots\!72}a^{20}+\frac{54\!\cdots\!53}{29\!\cdots\!76}a^{19}-\frac{27\!\cdots\!29}{21\!\cdots\!64}a^{18}-\frac{80\!\cdots\!53}{36\!\cdots\!72}a^{17}+\frac{36\!\cdots\!03}{36\!\cdots\!72}a^{16}-\frac{53\!\cdots\!91}{26\!\cdots\!16}a^{15}+\frac{35\!\cdots\!71}{13\!\cdots\!08}a^{14}-\frac{46\!\cdots\!71}{36\!\cdots\!72}a^{13}-\frac{25\!\cdots\!41}{92\!\cdots\!68}a^{12}+\frac{53\!\cdots\!13}{73\!\cdots\!44}a^{11}-\frac{16\!\cdots\!93}{18\!\cdots\!36}a^{10}+\frac{52\!\cdots\!45}{36\!\cdots\!72}a^{9}+\frac{82\!\cdots\!75}{92\!\cdots\!68}a^{8}-\frac{15\!\cdots\!23}{18\!\cdots\!36}a^{7}+\frac{35\!\cdots\!03}{28\!\cdots\!99}a^{6}+\frac{31\!\cdots\!57}{92\!\cdots\!68}a^{5}-\frac{22\!\cdots\!49}{11\!\cdots\!96}a^{4}+\frac{97\!\cdots\!69}{23\!\cdots\!92}a^{3}-\frac{26\!\cdots\!97}{23\!\cdots\!92}a^{2}-\frac{70\!\cdots\!31}{23\!\cdots\!92}a+\frac{23\!\cdots\!01}{57\!\cdots\!98}$, $\frac{34\!\cdots\!27}{26\!\cdots\!16}a^{31}-\frac{76\!\cdots\!21}{73\!\cdots\!44}a^{30}+\frac{30\!\cdots\!03}{36\!\cdots\!72}a^{29}-\frac{28\!\cdots\!93}{66\!\cdots\!04}a^{28}+\frac{23\!\cdots\!01}{11\!\cdots\!96}a^{27}-\frac{29\!\cdots\!13}{36\!\cdots\!72}a^{26}+\frac{38\!\cdots\!93}{13\!\cdots\!08}a^{25}-\frac{64\!\cdots\!43}{73\!\cdots\!44}a^{24}+\frac{88\!\cdots\!77}{36\!\cdots\!72}a^{23}-\frac{42\!\cdots\!21}{73\!\cdots\!44}a^{22}+\frac{89\!\cdots\!91}{73\!\cdots\!44}a^{21}-\frac{24\!\cdots\!71}{11\!\cdots\!96}a^{20}+\frac{81\!\cdots\!97}{29\!\cdots\!76}a^{19}-\frac{66\!\cdots\!63}{36\!\cdots\!72}a^{18}-\frac{12\!\cdots\!67}{36\!\cdots\!72}a^{17}+\frac{27\!\cdots\!23}{18\!\cdots\!36}a^{16}-\frac{81\!\cdots\!27}{26\!\cdots\!16}a^{15}+\frac{26\!\cdots\!57}{66\!\cdots\!04}a^{14}-\frac{14\!\cdots\!27}{83\!\cdots\!88}a^{13}-\frac{15\!\cdots\!61}{36\!\cdots\!72}a^{12}+\frac{81\!\cdots\!69}{73\!\cdots\!44}a^{11}-\frac{48\!\cdots\!93}{36\!\cdots\!72}a^{10}+\frac{61\!\cdots\!05}{36\!\cdots\!72}a^{9}+\frac{38\!\cdots\!31}{27\!\cdots\!08}a^{8}-\frac{22\!\cdots\!83}{18\!\cdots\!36}a^{7}-\frac{29\!\cdots\!71}{92\!\cdots\!68}a^{6}+\frac{47\!\cdots\!25}{92\!\cdots\!68}a^{5}-\frac{13\!\cdots\!97}{46\!\cdots\!84}a^{4}+\frac{12\!\cdots\!33}{23\!\cdots\!92}a^{3}-\frac{59\!\cdots\!63}{20\!\cdots\!72}a^{2}-\frac{66\!\cdots\!71}{23\!\cdots\!92}a-\frac{11\!\cdots\!69}{11\!\cdots\!96}$, $\frac{74\!\cdots\!63}{29\!\cdots\!76}a^{31}-\frac{30\!\cdots\!13}{14\!\cdots\!88}a^{30}+\frac{12\!\cdots\!15}{73\!\cdots\!44}a^{29}-\frac{39\!\cdots\!21}{46\!\cdots\!84}a^{28}+\frac{15\!\cdots\!51}{36\!\cdots\!72}a^{27}-\frac{60\!\cdots\!67}{36\!\cdots\!72}a^{26}+\frac{85\!\cdots\!37}{14\!\cdots\!88}a^{25}-\frac{66\!\cdots\!83}{36\!\cdots\!72}a^{24}+\frac{18\!\cdots\!61}{36\!\cdots\!72}a^{23}-\frac{89\!\cdots\!53}{73\!\cdots\!44}a^{22}+\frac{18\!\cdots\!99}{73\!\cdots\!44}a^{21}-\frac{16\!\cdots\!99}{36\!\cdots\!72}a^{20}+\frac{17\!\cdots\!19}{29\!\cdots\!76}a^{19}-\frac{65\!\cdots\!71}{14\!\cdots\!88}a^{18}-\frac{41\!\cdots\!65}{66\!\cdots\!04}a^{17}+\frac{22\!\cdots\!41}{73\!\cdots\!44}a^{16}-\frac{17\!\cdots\!21}{26\!\cdots\!16}a^{15}+\frac{11\!\cdots\!11}{13\!\cdots\!08}a^{14}-\frac{35\!\cdots\!21}{73\!\cdots\!44}a^{13}-\frac{57\!\cdots\!75}{73\!\cdots\!44}a^{12}+\frac{17\!\cdots\!67}{73\!\cdots\!44}a^{11}-\frac{27\!\cdots\!37}{92\!\cdots\!68}a^{10}+\frac{27\!\cdots\!75}{36\!\cdots\!72}a^{9}+\frac{62\!\cdots\!87}{23\!\cdots\!92}a^{8}-\frac{53\!\cdots\!37}{18\!\cdots\!36}a^{7}+\frac{14\!\cdots\!93}{46\!\cdots\!84}a^{6}+\frac{10\!\cdots\!35}{92\!\cdots\!68}a^{5}-\frac{39\!\cdots\!93}{52\!\cdots\!18}a^{4}+\frac{41\!\cdots\!63}{23\!\cdots\!92}a^{3}-\frac{82\!\cdots\!77}{23\!\cdots\!92}a^{2}-\frac{60\!\cdots\!23}{23\!\cdots\!92}a-\frac{12\!\cdots\!63}{11\!\cdots\!96}$, $\frac{16\!\cdots\!37}{13\!\cdots\!08}a^{31}-\frac{21\!\cdots\!23}{29\!\cdots\!76}a^{30}+\frac{87\!\cdots\!61}{14\!\cdots\!88}a^{29}-\frac{84\!\cdots\!97}{33\!\cdots\!52}a^{28}+\frac{10\!\cdots\!73}{92\!\cdots\!68}a^{27}-\frac{14\!\cdots\!11}{36\!\cdots\!72}a^{26}+\frac{85\!\cdots\!53}{66\!\cdots\!04}a^{25}-\frac{47\!\cdots\!81}{14\!\cdots\!88}a^{24}+\frac{27\!\cdots\!27}{36\!\cdots\!72}a^{23}-\frac{47\!\cdots\!35}{36\!\cdots\!72}a^{22}+\frac{10\!\cdots\!07}{73\!\cdots\!44}a^{21}+\frac{25\!\cdots\!41}{73\!\cdots\!44}a^{20}-\frac{12\!\cdots\!29}{14\!\cdots\!88}a^{19}+\frac{76\!\cdots\!09}{29\!\cdots\!76}a^{18}-\frac{81\!\cdots\!29}{14\!\cdots\!88}a^{17}+\frac{26\!\cdots\!01}{36\!\cdots\!72}a^{16}-\frac{24\!\cdots\!39}{13\!\cdots\!08}a^{15}-\frac{34\!\cdots\!71}{26\!\cdots\!16}a^{14}+\frac{59\!\cdots\!93}{13\!\cdots\!08}a^{13}-\frac{20\!\cdots\!61}{36\!\cdots\!72}a^{12}+\frac{13\!\cdots\!57}{73\!\cdots\!44}a^{11}+\frac{44\!\cdots\!93}{73\!\cdots\!44}a^{10}-\frac{16\!\cdots\!13}{92\!\cdots\!68}a^{9}+\frac{38\!\cdots\!25}{36\!\cdots\!72}a^{8}+\frac{25\!\cdots\!73}{18\!\cdots\!36}a^{7}-\frac{26\!\cdots\!71}{18\!\cdots\!36}a^{6}+\frac{13\!\cdots\!69}{23\!\cdots\!92}a^{5}+\frac{38\!\cdots\!97}{92\!\cdots\!68}a^{4}-\frac{11\!\cdots\!37}{46\!\cdots\!84}a^{3}+\frac{14\!\cdots\!29}{20\!\cdots\!72}a^{2}-\frac{18\!\cdots\!03}{23\!\cdots\!92}a+\frac{95\!\cdots\!89}{23\!\cdots\!92}$, $\frac{23\!\cdots\!65}{29\!\cdots\!76}a^{31}-\frac{47\!\cdots\!31}{73\!\cdots\!44}a^{30}+\frac{37\!\cdots\!87}{73\!\cdots\!44}a^{29}-\frac{39\!\cdots\!63}{14\!\cdots\!88}a^{28}+\frac{47\!\cdots\!07}{36\!\cdots\!72}a^{27}-\frac{37\!\cdots\!33}{73\!\cdots\!44}a^{26}+\frac{26\!\cdots\!23}{14\!\cdots\!88}a^{25}-\frac{40\!\cdots\!17}{73\!\cdots\!44}a^{24}+\frac{28\!\cdots\!81}{18\!\cdots\!36}a^{23}-\frac{33\!\cdots\!57}{92\!\cdots\!68}a^{22}+\frac{56\!\cdots\!01}{73\!\cdots\!44}a^{21}-\frac{13\!\cdots\!07}{10\!\cdots\!32}a^{20}+\frac{52\!\cdots\!41}{29\!\cdots\!76}a^{19}-\frac{44\!\cdots\!83}{36\!\cdots\!72}a^{18}-\frac{15\!\cdots\!69}{73\!\cdots\!44}a^{17}+\frac{14\!\cdots\!11}{14\!\cdots\!88}a^{16}-\frac{51\!\cdots\!43}{26\!\cdots\!16}a^{15}+\frac{85\!\cdots\!83}{33\!\cdots\!52}a^{14}-\frac{89\!\cdots\!05}{73\!\cdots\!44}a^{13}-\frac{38\!\cdots\!03}{14\!\cdots\!88}a^{12}+\frac{51\!\cdots\!47}{73\!\cdots\!44}a^{11}-\frac{62\!\cdots\!43}{73\!\cdots\!44}a^{10}+\frac{51\!\cdots\!15}{36\!\cdots\!72}a^{9}+\frac{16\!\cdots\!23}{18\!\cdots\!36}a^{8}-\frac{14\!\cdots\!65}{18\!\cdots\!36}a^{7}+\frac{20\!\cdots\!81}{18\!\cdots\!36}a^{6}+\frac{30\!\cdots\!03}{92\!\cdots\!68}a^{5}-\frac{87\!\cdots\!13}{46\!\cdots\!84}a^{4}+\frac{23\!\cdots\!77}{57\!\cdots\!98}a^{3}-\frac{52\!\cdots\!43}{46\!\cdots\!84}a^{2}-\frac{67\!\cdots\!67}{23\!\cdots\!92}a+\frac{10\!\cdots\!05}{23\!\cdots\!92}$, $\frac{63\!\cdots\!65}{80\!\cdots\!56}a^{31}-\frac{42\!\cdots\!07}{80\!\cdots\!56}a^{30}+\frac{42\!\cdots\!23}{10\!\cdots\!32}a^{29}-\frac{39\!\cdots\!09}{20\!\cdots\!64}a^{28}+\frac{18\!\cdots\!61}{20\!\cdots\!64}a^{27}-\frac{42\!\cdots\!97}{12\!\cdots\!04}a^{26}+\frac{46\!\cdots\!27}{40\!\cdots\!28}a^{25}-\frac{19\!\cdots\!53}{59\!\cdots\!84}a^{24}+\frac{25\!\cdots\!31}{29\!\cdots\!92}a^{23}-\frac{34\!\cdots\!37}{18\!\cdots\!24}a^{22}+\frac{34\!\cdots\!91}{10\!\cdots\!32}a^{21}-\frac{98\!\cdots\!33}{20\!\cdots\!64}a^{20}+\frac{26\!\cdots\!45}{80\!\cdots\!56}a^{19}+\frac{40\!\cdots\!77}{80\!\cdots\!56}a^{18}-\frac{56\!\cdots\!07}{20\!\cdots\!64}a^{17}+\frac{61\!\cdots\!01}{10\!\cdots\!32}a^{16}-\frac{55\!\cdots\!15}{72\!\cdots\!96}a^{15}+\frac{32\!\cdots\!81}{72\!\cdots\!96}a^{14}+\frac{61\!\cdots\!77}{50\!\cdots\!16}a^{13}-\frac{29\!\cdots\!27}{10\!\cdots\!32}a^{12}+\frac{60\!\cdots\!47}{20\!\cdots\!64}a^{11}-\frac{14\!\cdots\!51}{20\!\cdots\!64}a^{10}-\frac{62\!\cdots\!41}{10\!\cdots\!32}a^{9}+\frac{58\!\cdots\!43}{91\!\cdots\!12}a^{8}+\frac{16\!\cdots\!47}{50\!\cdots\!16}a^{7}-\frac{24\!\cdots\!35}{50\!\cdots\!16}a^{6}+\frac{12\!\cdots\!07}{25\!\cdots\!08}a^{5}+\frac{31\!\cdots\!29}{25\!\cdots\!08}a^{4}-\frac{16\!\cdots\!91}{31\!\cdots\!76}a^{3}+\frac{12\!\cdots\!39}{62\!\cdots\!52}a^{2}+\frac{51\!\cdots\!27}{62\!\cdots\!52}a-\frac{83\!\cdots\!35}{62\!\cdots\!52}$, $\frac{10\!\cdots\!65}{29\!\cdots\!76}a^{31}-\frac{79\!\cdots\!11}{29\!\cdots\!76}a^{30}+\frac{14\!\cdots\!43}{66\!\cdots\!04}a^{29}-\frac{20\!\cdots\!97}{18\!\cdots\!36}a^{28}+\frac{39\!\cdots\!09}{73\!\cdots\!44}a^{27}-\frac{17\!\cdots\!25}{83\!\cdots\!88}a^{26}+\frac{10\!\cdots\!27}{14\!\cdots\!88}a^{25}-\frac{33\!\cdots\!75}{14\!\cdots\!88}a^{24}+\frac{45\!\cdots\!11}{73\!\cdots\!44}a^{23}-\frac{10\!\cdots\!21}{73\!\cdots\!44}a^{22}+\frac{11\!\cdots\!11}{36\!\cdots\!72}a^{21}-\frac{39\!\cdots\!41}{73\!\cdots\!44}a^{20}+\frac{20\!\cdots\!77}{29\!\cdots\!76}a^{19}-\frac{12\!\cdots\!67}{29\!\cdots\!76}a^{18}-\frac{17\!\cdots\!87}{18\!\cdots\!36}a^{17}+\frac{26\!\cdots\!97}{66\!\cdots\!04}a^{16}-\frac{20\!\cdots\!63}{26\!\cdots\!16}a^{15}+\frac{26\!\cdots\!73}{26\!\cdots\!16}a^{14}-\frac{28\!\cdots\!19}{73\!\cdots\!44}a^{13}-\frac{86\!\cdots\!55}{73\!\cdots\!44}a^{12}+\frac{20\!\cdots\!51}{73\!\cdots\!44}a^{11}-\frac{23\!\cdots\!95}{73\!\cdots\!44}a^{10}+\frac{51\!\cdots\!93}{36\!\cdots\!72}a^{9}+\frac{13\!\cdots\!79}{36\!\cdots\!72}a^{8}-\frac{52\!\cdots\!57}{18\!\cdots\!36}a^{7}-\frac{67\!\cdots\!63}{18\!\cdots\!36}a^{6}+\frac{12\!\cdots\!97}{92\!\cdots\!68}a^{5}-\frac{57\!\cdots\!21}{92\!\cdots\!68}a^{4}+\frac{88\!\cdots\!53}{10\!\cdots\!36}a^{3}+\frac{17\!\cdots\!27}{23\!\cdots\!92}a^{2}+\frac{17\!\cdots\!43}{23\!\cdots\!92}a-\frac{62\!\cdots\!73}{20\!\cdots\!72}$, $\frac{10\!\cdots\!91}{29\!\cdots\!76}a^{31}-\frac{20\!\cdots\!25}{73\!\cdots\!44}a^{30}+\frac{80\!\cdots\!21}{36\!\cdots\!72}a^{29}-\frac{82\!\cdots\!49}{73\!\cdots\!44}a^{28}+\frac{49\!\cdots\!27}{92\!\cdots\!68}a^{27}-\frac{76\!\cdots\!79}{36\!\cdots\!72}a^{26}+\frac{10\!\cdots\!93}{14\!\cdots\!88}a^{25}-\frac{16\!\cdots\!69}{73\!\cdots\!44}a^{24}+\frac{22\!\cdots\!83}{36\!\cdots\!72}a^{23}-\frac{10\!\cdots\!63}{73\!\cdots\!44}a^{22}+\frac{22\!\cdots\!37}{73\!\cdots\!44}a^{21}-\frac{49\!\cdots\!99}{92\!\cdots\!68}a^{20}+\frac{20\!\cdots\!03}{29\!\cdots\!76}a^{19}-\frac{76\!\cdots\!39}{18\!\cdots\!36}a^{18}-\frac{35\!\cdots\!85}{36\!\cdots\!72}a^{17}+\frac{72\!\cdots\!19}{18\!\cdots\!36}a^{16}-\frac{20\!\cdots\!57}{26\!\cdots\!16}a^{15}+\frac{65\!\cdots\!41}{66\!\cdots\!04}a^{14}-\frac{34\!\cdots\!47}{92\!\cdots\!68}a^{13}-\frac{43\!\cdots\!55}{36\!\cdots\!72}a^{12}+\frac{20\!\cdots\!31}{73\!\cdots\!44}a^{11}-\frac{11\!\cdots\!55}{36\!\cdots\!72}a^{10}+\frac{45\!\cdots\!87}{36\!\cdots\!72}a^{9}+\frac{68\!\cdots\!67}{18\!\cdots\!36}a^{8}-\frac{51\!\cdots\!13}{18\!\cdots\!36}a^{7}-\frac{33\!\cdots\!95}{83\!\cdots\!88}a^{6}+\frac{12\!\cdots\!35}{92\!\cdots\!68}a^{5}-\frac{26\!\cdots\!75}{46\!\cdots\!84}a^{4}+\frac{21\!\cdots\!87}{23\!\cdots\!92}a^{3}-\frac{22\!\cdots\!03}{23\!\cdots\!92}a^{2}+\frac{11\!\cdots\!55}{23\!\cdots\!92}a-\frac{11\!\cdots\!15}{11\!\cdots\!96}$, $\frac{12\!\cdots\!89}{14\!\cdots\!88}a^{31}-\frac{10\!\cdots\!43}{14\!\cdots\!88}a^{30}+\frac{40\!\cdots\!99}{73\!\cdots\!44}a^{29}-\frac{52\!\cdots\!65}{18\!\cdots\!36}a^{28}+\frac{50\!\cdots\!45}{36\!\cdots\!72}a^{27}-\frac{98\!\cdots\!99}{18\!\cdots\!36}a^{26}+\frac{13\!\cdots\!51}{73\!\cdots\!44}a^{25}-\frac{42\!\cdots\!31}{73\!\cdots\!44}a^{24}+\frac{29\!\cdots\!91}{18\!\cdots\!36}a^{23}-\frac{14\!\cdots\!87}{36\!\cdots\!72}a^{22}+\frac{14\!\cdots\!95}{18\!\cdots\!36}a^{21}-\frac{51\!\cdots\!39}{36\!\cdots\!72}a^{20}+\frac{26\!\cdots\!45}{14\!\cdots\!88}a^{19}-\frac{16\!\cdots\!83}{14\!\cdots\!88}a^{18}-\frac{17\!\cdots\!59}{73\!\cdots\!44}a^{17}+\frac{37\!\cdots\!15}{36\!\cdots\!72}a^{16}-\frac{27\!\cdots\!75}{13\!\cdots\!08}a^{15}+\frac{34\!\cdots\!65}{13\!\cdots\!08}a^{14}-\frac{80\!\cdots\!35}{73\!\cdots\!44}a^{13}-\frac{10\!\cdots\!97}{36\!\cdots\!72}a^{12}+\frac{27\!\cdots\!79}{36\!\cdots\!72}a^{11}-\frac{28\!\cdots\!53}{33\!\cdots\!52}a^{10}+\frac{13\!\cdots\!29}{18\!\cdots\!36}a^{9}+\frac{17\!\cdots\!05}{18\!\cdots\!36}a^{8}-\frac{72\!\cdots\!29}{92\!\cdots\!68}a^{7}-\frac{75\!\cdots\!87}{92\!\cdots\!68}a^{6}+\frac{16\!\cdots\!23}{46\!\cdots\!84}a^{5}-\frac{76\!\cdots\!51}{46\!\cdots\!84}a^{4}+\frac{31\!\cdots\!51}{11\!\cdots\!96}a^{3}-\frac{63\!\cdots\!83}{11\!\cdots\!96}a^{2}-\frac{21\!\cdots\!97}{57\!\cdots\!98}a+\frac{23\!\cdots\!31}{11\!\cdots\!96}$, $\frac{28\!\cdots\!45}{29\!\cdots\!76}a^{31}-\frac{22\!\cdots\!61}{29\!\cdots\!76}a^{30}+\frac{90\!\cdots\!43}{14\!\cdots\!88}a^{29}-\frac{23\!\cdots\!13}{73\!\cdots\!44}a^{28}+\frac{57\!\cdots\!01}{36\!\cdots\!72}a^{27}-\frac{22\!\cdots\!47}{36\!\cdots\!72}a^{26}+\frac{32\!\cdots\!63}{14\!\cdots\!88}a^{25}-\frac{98\!\cdots\!37}{14\!\cdots\!88}a^{24}+\frac{34\!\cdots\!33}{18\!\cdots\!36}a^{23}-\frac{30\!\cdots\!95}{66\!\cdots\!04}a^{22}+\frac{34\!\cdots\!71}{36\!\cdots\!72}a^{21}-\frac{12\!\cdots\!77}{73\!\cdots\!44}a^{20}+\frac{65\!\cdots\!33}{29\!\cdots\!76}a^{19}-\frac{47\!\cdots\!97}{29\!\cdots\!76}a^{18}-\frac{34\!\cdots\!39}{14\!\cdots\!88}a^{17}+\frac{21\!\cdots\!87}{18\!\cdots\!36}a^{16}-\frac{64\!\cdots\!35}{26\!\cdots\!16}a^{15}+\frac{86\!\cdots\!95}{26\!\cdots\!16}a^{14}-\frac{25\!\cdots\!51}{14\!\cdots\!88}a^{13}-\frac{10\!\cdots\!33}{36\!\cdots\!72}a^{12}+\frac{15\!\cdots\!67}{18\!\cdots\!36}a^{11}-\frac{79\!\cdots\!13}{73\!\cdots\!44}a^{10}+\frac{94\!\cdots\!57}{36\!\cdots\!72}a^{9}+\frac{33\!\cdots\!33}{33\!\cdots\!52}a^{8}-\frac{47\!\cdots\!49}{46\!\cdots\!84}a^{7}+\frac{18\!\cdots\!19}{18\!\cdots\!36}a^{6}+\frac{34\!\cdots\!05}{92\!\cdots\!68}a^{5}-\frac{23\!\cdots\!73}{92\!\cdots\!68}a^{4}+\frac{32\!\cdots\!41}{46\!\cdots\!84}a^{3}-\frac{16\!\cdots\!87}{23\!\cdots\!92}a^{2}+\frac{52\!\cdots\!65}{57\!\cdots\!98}a+\frac{50\!\cdots\!71}{23\!\cdots\!92}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 14816072342282.787 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 14816072342282.787 \cdot 552}{6\cdot\sqrt{1119917091733566341117149537716450109685760000000000000000}}\cr\approx \mathstrut & 0.240328831332038 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 8*x^31 + 64*x^30 - 332*x^29 + 1604*x^28 - 6256*x^27 + 22310*x^26 - 68580*x^25 + 189272*x^24 - 457172*x^23 + 954788*x^22 - 1681104*x^21 + 2211865*x^20 - 1508956*x^19 - 2584912*x^18 + 11838608*x^17 - 24128969*x^16 + 31887592*x^15 - 15354096*x^14 - 32328344*x^13 + 86966400*x^12 - 105636528*x^11 + 18312240*x^10 + 107553536*x^9 - 98914352*x^8 + 2393408*x^7 + 40426432*x^6 - 24013312*x^5 + 5360768*x^4 - 207104*x^3 - 36096*x^2 + 1536*x + 256);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$D_4^2:C_2^3$ (as 32T12882):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1049\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |